The comments about "finite mathematics" are silly. "Finite mathematics" is not a field within mathematics, but rather a collection of diverse topics in elementary mathematics that the curriculum brings togetther in a single undergraduate course for business students. Truth tables are not different in "finite mathematics" than in other disciplines. -- Mike Hardy

The "arrow" connective, it is to be understood as a truth-functional operator, should not be described as "implication." Call it "conditional" or "if-then." Doing otherwise involves confusing the use-mention distinction that Quine first noticed and spent his whole career trying to enforce (Perhaps hopelessly: quantified modal logic is deeply infected with use-mention confusions.) See his Mathematical Logic, Section 5.

In any case, "if...then" is not the same as "implies." "Implies" is a relation between sentences: a two-place predicate that takes sentences as the values of its variables and produces a sentence from them: it is a function from names of sentences--terms--to a sentence.

By constrast "if then" is a not a predicate but a connective; it is a funtion from sentences to a sentence. It does not take anything as values because it does not contain variables.

"Implies" talks about--mentions--two sentences, and can only be used in a meta-language. "If...then" uses two sentences; it mentions whatever the sentences mention, and is itself a term within the object language. Shortly:

If A then B. If the light goes out then the monsters will come.

but

"A" implies "B". "The light goes out" implies "The monsters will come".

Sorry for the rant. If anyone sees this mistake elsewhere, please correct it.

(Added) Same goes for equivalence. Sentences are equivalent to one another; but the things they say are related as "...if and only if ..."

Also, variables don't generally have truth-values. That's too confused to explain at all. P, Q, and R here aren't being used as variables, (though if they were, they'd have sentences, not truth-values, as their values). They're schemata; their standing in for sentences, but there's no assumption that you can quantify over them. The best way to explain this stuff is using Quine-corners and his Greek-letter metavariables. But, alas, no one cares about being rigorous anymore. Sigh.

Peirce

Can anyone provide a reference for Charles Peirce’s development of truth tables? I would like to check the form that they took. It appears from a quick bit of research that the tables he developed were substantially different in form to those presented here (see http://plato.stanford.edu/entries/peirce-logic/ ) Those sources that support the claim appear to derive from the Wikipedia. Was it Wittgenstein who developed the form that is now used? Banno 22:59, May 15, 2004 (UTC)

Discussed in depth in this thread: http://sunsite.utk.edu/math_archives/.http/hypermail/historia/apr00/0117.html

An excellent summary of the issue. Thanks for pointing it out to me. The conclusion appears to be that Frege, Peirce, and Schröder all played a part in the development of the truth table, and so the attribution of them to Peirce alone should be altered. Wittgenstein perhaps had the role of popularising their use. Banno 00:04, May 16, 2004 (UTC)